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The clearest of the Higher papers

  1. Jun 25, 2006 #1

    marcus

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    the clearest of the "Higher" papers

    I came to an odd conclusion as I was trying to read three papers and see where they were going---I'll toss it out in case anybody might be able to elaborate, or have a different idea

    It is that the clearest of the papers is the 2005 one "Higher Gauge Theory" of Baez and Schreiber. This is the one that, for me, TELLS THE STORY.

    or tells it the most simply, in the fewest pages, up front. It says why twogroups.

    All these "Higher" papers are thin and downloadable. You can easily print them off and they are short enough to read without having to worry about what to skip. What I am saying is that the BaezSchreiber one is the one that works best JOURNALISTICALLY.

    As you read it you can know for certain why you are. I will try to tell the story in the next post. It may take a day or so (the next realworld day is very busy for me) and anybody who wants can take over.
     
    Last edited: Jun 25, 2006
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  3. Jun 25, 2006 #2

    marcus

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    Some "Higher" papers are:
    http://arxiv.org/abs/math.DG/0511710
    Higher Gauge Theory
    John C. Baez, Urs Schreiber
    28 pages, 10 figures

    "Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings) using 2-connections on 2-bundles. A 2-bundle is a categorified version of a bundle: that is, one where the fiber is not a manifold but a category with a suitable smooth structure. Where gauge theory uses Lie groups and Lie algebras, higher gauge theory uses their categorified analogues: Lie 2-groups and Lie 2-algebras. We describe a theory of 2-connections on principal 2-bundles and explain how this is related to Breen and Messing's theory of connections on nonabelian gerbes. The distinctive feature of our theory is that a 2-connection allows parallel transport along paths and surfaces in a parametrization-independent way. In terms of Breen and Messing's framework, this requires that the "fake curvature" must vanish. In this paper we summarize the main results of our theory without proofs.

    http://arxiv.org/abs/hep-th/0206130
    Higher Yang-Mills Theory
    John C. Baez
    20 pages

    And there is the Perimeter lecture where the video is online and also the slides (which serve as lecture notes)
    http://math.ucr.edu/home/baez/quantum_spacetime/
    Higher-Dimensional Algebra: A Language for Quantum Spacetime

    and this page has all the links! I actually dont have to be writing any links----for example the 2005 Baez Schreiber H.G.T paper is here.

    OK, now I have to say why I think that particular paper tells the story in the clearest way-----why stepping up to 2-connections is necessary and remarkably how it actually explains why BF theory. (alias the vanishing of the fake curvature).

    I have to try to retell the story. But, it looks like the realworld day is (not unwelcomely in this case) crowding in around us and I wont be able to start for several hours.
    so unless a sliver of time appears this will just have to sit. feel free to jump in
     
    Last edited: Jun 25, 2006
  4. Jun 25, 2006 #3

    marcus

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    sliver: my basic attitude is that quantizing spacetime means its geometry and (from within) geometry is about what happens to things as you move a point along a curve----you never get OUTSIDE your geometry looking down on it, so you have to picture exploring and discovering it by moving things and seeing how they change:

    i.e. what happens to a particle, or a gyroscope, or someone's POV, as a point moves along a curve

    and "what happens" traditionally means a GROUP

    so representing an instantaneous state of geometry by a SPIN NETWORK makes sense---it is just a web of CURVES along which you can move POINTS with the bare minimum of machinery so that a NUMBER can be extracted from the GROUP element you get by moving a point along and seeing how things change.

    and to a lot of people, including myself, the obvious way to represent the way a spin network evolves in time is by a honeycomb contraption called a spin foam in which there are flat pieces as well as lines, or bits of surface as well as bits of curve. A flat piece just shows how a curve evolved in time. a spinfoam is just what a spin network sweeps out if you drag it thru time.

    and something has bothered me for several years about spinfoams:

    Remember it is made of line segments and flats, or curves and sections of surface. I can understand moving a point along a curve and having something change, and getting a group element (the basic "geometrical act" in a way). that much is OK.

    But I always had trouble picturing how you could MOVE A CURVE ACROSS A BIT OF SURFACE and get a group element. And would it even be an element of the same group!?

    so I could easily understand the labeling of spin networks (and the instantaneous state of spatial geomentry) but I could not understand the labeling of the bits of surface in the spin foam.

    this is the first thing that you get made very clear in the BaezSchreiber HGT paper.

    and one of the nice things is they show you very simply WHY you cant use the same GROUP for two jobs. You can't use the same group BOTH to track the changes as you snake the point along the curve, AND track the changes as yuou snake the curve across the surface! If you try to be cheap, and get away with using the same group to clock the changes, then THE DAMNED GROUP will be FORCED TO BE ABELIAN. Overworking the group makes it turn out to be commutative---which is uncool.

    they actually prove this with a few very simple pictures in the first PAGE AND A HALF or so.

    I have to go now. I will get back to this later. But I may as well jump ahead and say that this business of tracking changes is called holonomy----or sometimes a connection. those words are floating around.
    and they actually do construct a holonomy setup that HANDLES the moving points along curves AND snaking the curves across the surfaces----and they show that you need a TWOGROUP to do it.

    that in one blow justifies learning about twogroups, for me, because I was always worried about the meaning of holonomy in spinfoams. you need the twogroups because you cant do it some other way, why, it almost seems natural!

    and then the paper also says why BEEF! which is a real bonanza.

    the holonomy setup, to work properly, has to be set up so it has no "fake curvature" and if you set the fake curvature to zero you get an equation resembling the beef equation. this amusing concept of fake curvature was made up by breen and messing

    ooops have to go
    ==============
    [EDIT] may be able to organize this better tomorrow if I take a second stab at it. there is a earlier "Higher" paper that has more complete proofs. the short 20-30 page paper seems to represent the cream skimmed off of this one, which I will list for completeness:

    http://arxiv.org/abs/hep-th/0412325
    Higher Gauge Theory: 2-Connections on 2-Bundles
    John Baez, Urs Schreiber
    73 pages, 4 figures
    "Connections and curvings on gerbes are beginning to play a vital role in differential geometry and mathematical physics -- first abelian gerbes, and more recently nonabelian gerbes. These concepts can be elegantly understood using the concept of '2-bundle' recently introduced by Bartels. A 2-bundle is a generalization of a bundle in which the fibers are categories rather than sets. Here we introduce the concept of a '2-connection' on a principal 2-bundle. We describe principal 2-bundles with connection in terms of local data, and show that under certain conditions this reduces to the cocycle data for nonabelian gerbes with connection and curving subject to a certain constraint -- namely, the vanishing of the 'fake curvature', as defined by Breen and Messing. This constraint also turns out to guarantee the existence of '2-holonomies': that is, parallel transport over both curves and surfaces, fitting together to define a 2-functor from the `path 2-groupoid' of the base space to the structure 2-group. We give a general theory of 2-holonomies and show how they are related to ordinary parallel transport on the path space of the base manifold."

    an important word of advice is don't be put off just because someone says "gerbe"
    mathematicians can't help themselves and wills say that kind of thing now and then. what matters is
    "...guarantee the existence of '2-holonomies': that is, parallel transport over both curves and surfaces"

    ==========
    language note: "holos" means the ALL, "nomos" means LAW, holonomy means FINDING OUT THE OVERALL PICTURE ("the law of the all") by cruising around. since you cant get outside your geometry and look down on it you have to explore it by cruising and seeing how things (like particles, gyroscopes, etc.) change. I think maybe submarines under the ocean do this, holonomy is not all that weird or unusual.

    gerbe, I am told, means spinach. the message being that a respected French geometry expert encourages us to eat our spinach. However I could be wrong. It might mean "sheaf" :smile:
     
    Last edited: Jun 25, 2006
  5. Jun 25, 2006 #4

    marcus

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    remember that one way to understand how a twogroup looks and acts is to think of it as a "crossmodule" COMBO of two ordinary onegroups G and H.

    there really is a way to take two usual groups G and H with some reciprocal linkage and combine them into a twogroup!

    one very nice thing is that when you have a two-holonomy setup where a connection is telling about PARALLEL TRANSPORT of stuff over curves AND over surfaces then guess what happens?

    transport a point along a curve and you get an element of G

    transport a curve across a surface and you get an element of H!

    so each of the component groups (that together merge to form the twogroup) is doing something essential and important.

    and you can be sure about this because if you try to get away with just using one group for both things then, JB and Urs show you in the first page-and-half, that it forces the group to be Abelian-----overworking the group makes it go down to pedestrian humiliation.
    to me this looks like sufficient reason why twogroups and why two-holonomy with two-connections. the other kind simply doesnt make it
    ==============
    Argh, I thought I was going to be able to get away with not printing out that long 2004 BaezSchreiber paper but it is looking more and more like I will need to print it out. It covers much the same topics as the nice short "Higher Gauge Theory" paper by the same authors----but has more proofs and more details. So it is the slow motion treatment of H.G.T. and might be handy to be able to consult.
    http://uk.arxiv.org/pdf/hep-th/0412325
    Higher Gauge Theory: 2-Connections on 2-Bundles
    John Baez, Urs Schreiber
    73 pages, 4 figures
    I already quoted the abstract

    just as a sample exerpt, here is the first thing they say in the introduction:
    "1. Introduction

    The gauge principle and the concept of connection is at the very heart of modern physics, and is also central to much of modern mathematics. It is all about parallel transport along curves. Due to the influence of string theory on the one hand (see §1.2 (p.7)) and higher category theory on the other (see §2.1 (p.9)), there are compelling reasons to generalize this concept to higher dimensions and find a notion of parallel transport along surfaces. "

    Interesting that they don't mention non-string QG here and in particular don't mention spinfoam. I could be fooling myself about the relevance of 2-holonomy to spinfoam, but that was what got me interested---and was the message i got out of the notes to Baez perimeter talk
     
    Last edited: Jun 26, 2006
  6. Jun 26, 2006 #5

    john baez

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    higher gauge theory papers

    That's nice to hear! And the reason it's nice to hear is that this paper is supposed to tell the story of higher gauge theory thus far. It's a review article that should appear in the proceedings of the Streetfest - the big conference in honor of Ross Street's 60th birthday.

    My paper "Higher Yang-Mills Theory" got the ball rolling, but there are things about it I don't like, so I'm letting it roll into oblivion.

    My first real papers on higher gauge theory are HDA5: 2-Groups with Aaron Lauda and HDA6: Lie 2-Algebras with Alissa Crans. These are mainly for mathematicians: we laid down some of the basic infrastructure of higher gauge theory - categorified groups and Lie algebras - without saying much about applications.

    Then comes From Loop Groups to 2-Groups, by Alissa, Danny Stevenson, Urs Schreiber and myself. Here we solved a problem raised in HDA6, namely to find the 2-groups corresponding to certain Lie 2-algebras that naturally arise from the math of string theory.

    Next - in logical but not chronological order - comes Toby Bartels' paper Higher Gauge Theory I: 2-bundles. He's a student of mine, and this is his thesis; he just finished it a week or two ago!

    Next will come a paper I wrote with Urs, Higher Gauge Theory II: 2-connections. This isn't even done yet: the version on my website is one step closer to the final product than the version currently on the arXiv, which you cite (and which currently has a slightly different title). I hope I can finish it while I'm hanging out in Shanghai this summer.

    Next - in logical order - comes Urs' thesis, From Loop Space Mechanics to Nonabelian Strings. Here he applies all the previous stuff to string theory.

    It's been a lot of work writing these papers, and I have two more 60-page papers I need to finish on n-categories in topology and physics... it's all very tiring - 1% inspiration and 99% perspiration. :frown:

    But, if I ever have the energy, an obvious next step is to understand the M-theory 3-supergroup and how the corresponding 3-connections relate to 11-dimensional supergravity. That would be lots of fun. I hope I can do it with Urs Schreiber, and maybe a grad student.

    Excellent - that's nice to hear.
     
  7. Jun 26, 2006 #6

    marcus

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    I am glad you point out that this
    http://math.ucr.edu/home/baez/2conn.pdf
    is a newer version. Since it is around 74 pages, if I print it out it will be good to have the latest.

    My eye was caught by a reference to the non-Abelian Stokes theorem on page 66 with reference [56] to a paper of Mansouri and others.

    It looked to me as if this was saying the same as what I earlier understood as requiring "fake curvature" to vanish. Also a connection to the equation of BF.

    it would be nice. I would like to think of this condition as a "nonAbelian Stokes theorem":cool:

    the same thing in the version that currently on arxiv is at page 52. and the same Mansouri reference is only number [39]----so you have been busy---the paper has grown considerably

    any words on the application of H.G.T. to spinfoam, or non-string QG generally? work in progress?
    I think this may relate to work you mentioned of Baratin and Freidel.
    ==========

    what you said about this in the Perimeter talk (page 11 of the slides) was interesting:

    "Both string theory and spin foam models are trying to exploit this clue. they are groping towards a language for quantum spacetime that will usefully blur the distinction between pieces of spacetime geometry and quantum processes.
    At this point we should think of them, not as predictive theories, but as explorations of the mathematical possibilities!"

    that is a striking observation but leaves me wanting to know some elaboration in terms of current work in spinfoam.
    please forgive the unbridled curiosity
     
    Last edited: Jun 27, 2006
  8. Jun 27, 2006 #7

    marcus

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    I suppose everybody benefits if JB does NOT reply to questions on a thread like this but instead puts time into the papers he mentioned he is writing. On the other hand I have questions which I want to air. Maybe SOMEBODY ELSE can answer, in any case JB will hopefully not feel obliged to. Maybe the important thing is to formulate the questions, whether or not anyone replies.

    What I think I will do is TRANSCRIBE the last couple of pages of the JB perimeter talk "Higher...Language for Quantum Spacetime" where he lays out FOUR POSSIBLE APPLICATIONS. there is just a couple of sentences on each of the four. what I want is clarification and elaboration on each of those, especially #2 and #4.

    Maybe I will emphasize #2 and #4 and not type out the other so much.
     
  9. Jun 27, 2006 #8

    marcus

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    the following is partially paraphrased and eccentrically repunctuated, from the last two pages or so.

    ==exerpts paraphrased from JB Perimeter talk==
    WE SEE TWOCATEGORIES PLAYING FOUR DISTINCT BUT RELATED ROLES

    #1. In string theory---more precisely in ANY CONFORMAL FIELD THEORY---we have a twocategory where:

    objects are D-branes
    morphisms are string states
    twomorphisms are evolution operators corresp. to string worldsheets
    (Runkel, Fuchs, Schweigert)
    (my question: hey! even without extra dimensions! what happens when you do string theory with new tools and no extra dimensions? Do Runkel Fuchs and Schweigert tell us?)

    #2 In ANY EXTENDED TOPOLOGICAL QUANTUM FIELD THEORY we have a twocategory where
    objects are matter
    morphisms describe choices of space
    twomorphisms describe choices of spacetime
    and Freidel et al have worked on the 3D case and
    in 4D the MATTER CONSISTS OF STRINGS (see the Loop Braid papers of Baez, Crans, Wise, Perez)
    (my question: this is background independent string theory with no extra dimensions and something like matter happens! is this a foretaste of the anticipated Freidel et al papers, with Baratin and with Starodubtsev? does Freidel get into having WRIGGLERS in his 4D or 5D world?)

    #3 In HIGHER GAUGE THEORY we have fields describing parallel transport not just for points moving along curves but also for strings tracing out surfaces. (Baez, Bartels, Crans, Lauda, Schreiber, Stevenson)
    And this is the stuff that is so far most clearly and explicitly DEVELOPED IN A SERIES OF PAPERS which JB has been discussing here happily for us, so I skip the paraphrase and go to the last sentence which is
    "SO WE GET PARALLEL TRANSPORT FOR PARTICLES AND STRINGS IN 4D TOPOLOGICAL GRAVITY!"
    (my comment: no extra dimensions at least in the last sentence. JB's italics and exclamation point. looks like Higher Gauge Theory is the queen on the chessboard here, or a pawn about to be promoted)

    #4 Feynman diagrams and spin networks are REALLY JUST WAYS OF REASONING DIAGRAMMATICALLY WITH OPERATORS-MORPHISMS IN HILB.

    (comment: one would think one ought really take that to heart)

    Similarly, spin foams are a way of reasoning with twomorphisms in a twocategory of 'TWOHILBERTSPACES'.

    (comment: OK please tell us about twoHilberts! Certainly the original Hilbertspace was a fertile invention. If twoHilberts are even remotely as useful, it would be nice to be introduced.)
    ===end of paraphrase===

    Did JB already discuss twoHilberts in one of those longer papers that I shrink from printing out for fear of total bafflement? Shall we at PF attempt to INVENT twoHilberts?
    That paper called "Quantum Quandaries" was fun to read and short and not too hard. I don't remember anything about twoHilberts in it though.
    Can anyone help out here?
     
    Last edited: Jun 27, 2006
  10. Jun 27, 2006 #9

    john baez

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    2-Hilbert spaces

    Higher-dimensional algebra II: 2-Hilbert spaces
    http://arxiv.org/abs/q-alg/9609018

    I explained the basic idea in week99 of This Week's Finds.

    Here's what I wrote... but you like punchlines, perhaps I should start
    with the punchline:

    When physicists do Feynman path integration - just like a shepherd
    counting sheep - they are engaged in a process of decategorification!


    They get a mere number saying the amplitude for something to happen,
    when they could get a Hilbert space of ways for things to happen.
    This amplitude is the inner product in a Hilbert space; the Hilbert space
    they could have gotten would be the inner product in a 2-Hilbert space.

    Mindblowing, huh?

    In other words: the complex numbers are just a shrunken,
    miniaturized version of the category of Hilbert spaces, and similarly
    the category of Hilbert spaces is just a shrunken, miniaturized version
    of the 2-category of 2-Hilbert spaces... and so on, ad infinitum. This
    "shrinking down" is called decategorification: it happens whenever
    you pretend isomorphic things are equal. I believe that to do math and
    physics wisely, we need to undo this pretense.

    Anyway, here's what I wrote:


    I want to say a bit about what category theory has to do with quantum
    mechanics!

    First remember the big picture: n-category theory is a language to
    talk about processes that turn processes into other processes.
    Roughly speaking, an n-category is some sort of structure with
    objects, morphisms between objects, 2-morphisms between morphisms,
    and so on up to n-morphisms. A 0-category is just a set, with its objects
    usually being called "elements". Things get trickier as n increases.
    For a precise definition of n-categories for n = 1 and 2, see "week73"
    and "week80", respectively.

    Most familiar mathematical gadgets are sets equipped with extra bells
    and whistles: groups, vector spaces, Hilbert spaces, and so on all
    have underlying sets. This is why set theory plays an important role
    in mathematics. However, we can also consider fancier gadgets that
    are *categories* equipped with extra bells and whistles. Some of the
    most interesting examples are just "categorifications" of well-known
    gadgets.

    For example, a "monoid" is a simple gadget, just a set equipped with
    an associative product and multiplicative identity. An example we all
    know and love is the complex numbers: the product is just ordinary
    multiplication, and the multiplicative identity is the number 1.

    We may categorify the notion of "monoid" and define a "monoidal
    category" to be a *category* equipped with an associative product and
    multiplicative identity. I gave the precise definition back in
    "week89"; the point here is that while they may sound scary, monoidal
    categories are actually very familiar. For example, the category of
    Hilbert spaces is a monoidal category where the product of Hilbert
    spaces is the tensor product and the multiplicative identity is C, the
    complex numbers.

    If one systematically studies categorification one discovers an
    amazing fact: many deep-sounding results in mathematics are just
    categorifications of stuff we all learned in high school. There is a
    good reason for this, I believe. All along, mathematicians have been
    unwittingly "decategorifying" mathematics by pretending that
    categories are just sets. We "decategorify" a category by forgetting
    about the morphisms and pretending that isomorphic objects are equal.
    We are left with a mere set: the set of isomorphism classes of
    objects.

    I gave an example in "week73". There is a category FinSet whose
    objects are finite sets and whose morphisms are functions. If we
    decategorify this, we get the set of natural numbers! Why? Well, two
    finite sets are isomorphic if they have the same number of elements.
    "Counting" is thus the primordial example of decategorification.

    I like to think of it in terms of the following fairy tale. Long ago, if
    you were a shepherd and wanted to see if two finite sets of sheep were
    isomorphic, the most obvious way would be to look for an isomorphism.
    In other words, you would try to match each sheep in herd A with a
    sheep in herd B. But one day, along came a shepherd who invented
    decategorification. This person realized you could take each set and
    "count" it, setting up an isomorphism between it and some set of
    "numbers", which were nonsense words like "one, two, three, four,..."
    specially designed for this purpose. By comparing the resulting
    numbers, you could see if two herds were isomorphic without explicitly
    establishing an isomorphism!

    According to this fairy tale, decategorification started out as the
    ultimate stroke of mathematical genius. Only later did it become a
    matter of dumb habit, which we are now struggling to overcome through
    the process of "categorification".

    Okay, so what does this have to do with quantum mechanics?

    Well, a Hilbert space is a set with extra bells and whistles, so maybe
    there is some gadget called a "2-Hilbert space" which is a *category*
    with analogous extra bells and whistles. And maybe if we figure out
    this notion we will learn something about quantum mechanics.

    Actually the notion of 2-Hilbert space didn't arise in this
    simple-minded way. It arose in some work by Daniel Freed on
    topological quantum field theory:

    5) Higher algebraic structures and quantization, by Dan Freed,
    Comm. Math. Phys. 159 (1994), 343-398, preprint available as
    hep-th/9212115; see also week48.

    Later, Louis Crane adopted this notion as part of his program to
    reduce quantum gravity to n-category theory:

    6) Louis Crane: Clock and category: is quantum gravity algebraic?,
    Jour. Math. Phys. 36 (1995), 6180-6193, preprint available as
    gr-qc/9504038.

    These papers made is clear that 2-Hilbert spaces are interesting
    things and that one should go further and think about "n-Hilbert
    spaces" for higher n, too. However, neither of them gave a precise
    definition of 2-Hilbert space, so at some point I decided to do this.
    It took a while for me to learn enough category theory, but eventually
    I wrote something about it:

    7) John Baez, Higher-dimensional algebra II: 2-Hilbert spaces,
    to appear in Adv. Math., available at q-alg/9609018.

    To understand this requires a little category theory, so let
    me explain the basic ideas here.

    I'll concentrate on finite-dimensional Hilbert spaces, since the
    infinite-dimensional case introduces extra complications. To define
    2-Hilbert spaces we need to start by categorifying the various
    ingredients in the definition of Hilbert space. These are: 1) the
    zero element, 2) addition, 3) subtraction, 4) scalar multiplication,
    and 5) the inner product. The first four have well-known categorical
    analogs. The fifth one, which is really the essence of a Hilbert
    space, may seem a bit more mysterious at first, but as we shall see,
    it's really the key to the whole business.

    1) The analog of the zero vector is a `zero object'. A zero object in
    a category is an object that is both initial and terminal. That is,
    there is exactly one morphism from it to any object, and exactly one
    morphism to it from any object. Consider for example the category
    Hilb having finite-dimensional Hilbert spaces as objects, and linear
    maps between them as morphisms. In Hilb, any zero-dimensional Hilbert
    space is a zero object.

    Note: there isn't really a unique zero object in the "strict" sense of
    the term. Instead, any two zero objects are canonically isomorphic.
    The reason is that if you have two zero objects, say 0 and 0', there
    is a unique morphism f: 0 -> 0' and a unique morphism g: 0' -> 0.
    These morphisms are inverses of each other so they are isomorphisms.
    Why are they inverses? Well, fg: 0 -> 0' must be the identity
    morphism 1_0: 0 -> 0, because there is only one morphism from 0 to 0!
    Similarly, gf is the identity on 0'. (Note that I am using category
    theorist's notation, where the composite of f: x -> y and g: y -> z is
    denoted fg: x -> z.)

    This is typical in category theory. We don't expect stuff to be
    unique; it should only be unique up to a canonical isomorphism.

    2) The analog of adding two vectors is forming the "coproduct" of two
    objects. Coproducts are just a fancy way of talking about direct
    sums. Any decent quantum mechanic knows about the direct sum of
    Hilbert spaces. But in fact, we can define this notion very generally
    in any category, where it goes under the name of a "coproduct". (I
    give the definition below; if I gave it here it would scare people
    away.) As with zero objects, coproducts are typically not unique, but
    they are always unique up to canonical isomorphism, which is what
    matters. It's a good little exercise to show this.

    3) The analog of subtracting vectors is forming the "cokernel" of a
    morphism f: x -> y. If x and y are Hilbert spaces, the cokernel of f
    is just the orthogonal complement of the range of f. In other words,
    for Hilbert spaces we have "direct differences" as well as direct
    sums. However, the notion of cokernel makes sense in any category
    with a zero object. I won't burden you with the precise definition
    here.

    An important difference between zero, addition and subtraction and
    their categorical analogs is that these operations represent extra
    *structure* on a set, while having a zero object, coproducts of two
    objects, or cokernels of morphisms is merely a *property* of a
    category. Thus these concepts are in some sense more intrinsic to
    categories than to sets. On the other hand, we've seen one pays a
    price for this: while the zero element, sums, and differences are
    unique in a Hilbert space, the zero object, coproducts, and cokernels
    are typically unique only up to canonical isomorphism.

    4) The analog of multiplying a vector by a complex number is tensoring
    an object by a Hilbert space. Besides its additive properties (zero
    object, binary coproducts, and cokernels), Hilb is also a monoidal
    category: we can multiply Hilbert space by tensoring them, and there
    is a multiplicative identity, namely the complex numbers C. In
    fact, Hilb is a "ring category", as defined by Laplaza and Kelly.

    We expect Hilb it to play a role in 2-Hilbert space theory analogous
    to the role played by the ring C of complex numbers in Hilbert space
    theory. Thus we expect 2-Hilbert spaces to be "module categories"
    over Hilb, as defined by Kapranov and Voevodsky.

    An important part of our philosophy here is that C is the primordial
    Hilbert space: the simplest one, upon which the rest are modelled. By
    analogy, we expect Hilb to be the primordial 2-Hilbert space. This is
    part of a general pattern pervading higher-dimensional algebra; for
    example, there is a sense in which the (n+1)-category of all (small)
    n-categories, nCat, is the primordial (n+1)-category. The real
    significance of this pattern remains mysterious.

    5) Finally, what is the categorification of the inner product in a
    Hilbert space? It's the `hom functor'! The inner product in a
    Hilbert space eats two vectors v and w and spits out a complex number

    <v,w>

    Similarly, given two objects v and w in a category, the hom functor
    gives a *set*

    hom(x,y)

    namely the set of morphisms from x to y. Note that the inner product
    <v,w> is linear in w and conjugate-linear in y, and similarly, the hom
    functor hom(x,y) is covariant in y and contravariant in x. This hints
    at the category theory secretly underlying quantum mechanics. In
    quantum theory the inner product <v,w> represents the *amplitude* to
    pass from v to w, while in category theory hom(x,y) is the *set* of
    ways to get from x to y. In Feynman path integrals, we do an integral
    over the set of ways to get from here to there, and get a number, the
    amplitude to get from here to there. So when physicists do Feynman
    path integration - just like a shepherd counting sheep - they are engaged
    in a process of decategorification!

    To understand this analogy better, note that any morphism f: x -> y in
    Hilb can be turned around or "dualized" to obtain a morphism f*: y -> x.
    This is usually called the "adjoint" of f, and it satisfies

    <fv,w> = <v,f*w>

    for all v in x, and w in y. This ability to dualize morphisms is
    crucial to quantum theory. For example, observables are represented
    by self-adjoint morphisms, while symmetries are represented by unitary
    morphisms, whose adjoint equals their inverse.

    However, it should now be clear - at least to the categorically minded -
    that this sort of adjoint is just a decategorified version of the
    "adjoint functors" so important in category theory. As I explained in
    "week79", a functor F*: D -> C is a "right adjoint" of F: C -> D if
    there is, not an equation, but a natural isomorphism

    hom(Fc,d) ~ hom(c,F*d)

    for all objects c in C, and d in D.

    Anyway, in the paper I proceed to use these ideas to give a precise
    definition of 2-Hilbert spaces, and then I prove all sorts of stuff
    which I won't describe here.

    Let me wrap up by explaining the definition of "coproduct". This is
    one of those things they should teach all math grad students, but for
    some reason they don't. It's a bit dry but it'll be good for you. A
    coproduct of the objects x and y is an object x+y equipped with
    morphisms

    i: x -> x+y

    and

    j: y -> x+y,

    that is universal with respect to this property. In other words,
    if we have any *other* object, say z, and morphisms

    i': x -> z

    and

    j': y -> z,

    then there is a unique morphism f: x+y -> z such that

    i' = if

    and

    j' = jf.

    This kind of definition automatically implies that the coproduct is
    unique up to canonical isomorphism. To understand this abstract
    nonsense, it's good to check that the coproduct of sets or topological
    spaces is just their disjoint union, while the coproduct of vector
    spaces or Hilbert spaces is their direct sum.
     
  11. Jun 27, 2006 #10

    john baez

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    Yes, sorry, it keeps getting longer. I still need to go through the stuff at the end and whip it into shape; not all the notation is consistent, etc. etc.

    The nonabelian Stokes theorem is a theorem; the fake flatness condition is a condition, so they're not "the same" - but they are indeed closely related, and closely related to the basic equation of BF theory.

    The nonabelian Stokes theorem says how to express the integral of a Lie-algebra-valued 1-form around a loop in terms of an integral over a surface having that loop as boundary. A lot like the Stokes theorem we know and love in electromagnetism, but now for more general gauge fields! It's more complicated, and for some idiotic reason it's not in any textbooks - not even mine :uhh: - so people keep rediscovering it....

    In higher gauge theory we need the holonomy of the Lie-algebra-valued 1-form A along curves to be related to the holonomy of the Lie-algebra-valued 2-form B over surfaces to get things to work nicely. So, we need a condition relating A and B: the "fake flatness condition". And, to show this condition suffices, we need to use the nonabelian Stokes theorem.

    The basic equation of BF theory says that B is proportional to F, the curvature of A. This is a special case of the fake flatness condition.


    It's all part of a big picture, but unfortunately nobody has enough patience to think deeply about this big picture, so they go chewing around the edges...

    One reason I like higher gauge theory is that it seems applicable both to string theory and also - possibly - loop quantum gravity. The reason is simple: both strings and loops are 1-dimensional entities that move through spacetime tracing out 2-dimensional surface: "string worldsheets" or "spin foams".

    Ordinary gauge theory handles the motion of 0-dimensional entities that move through spacetime tracing out 1-dimensional curves; maybe we should go up one notch! If we do, we get the simplest case of higher gauge theory.

    In that mysterious mess called "M-theory", we also hints of still higher-dimensional membranes moving through spacetime, and treating these could call for still higher kinds of gauge theory.

    As a mathematician, what I love is that these higher-dimensional surfaces look like the diagrams people draw in n-category theory: an "n-morphism" looks like an n-dimensional surface, and we can think of it as a "process between processes between processes...." So, it's not surprising that n-category theory is the right math for higher gauge theory - but it's a
    lot of fun working out the details!

    For example, I mentioned recently here, ordinary gauge theory describing the motion of point particles uses groups, and higher gauge theory for the motion of loops or strings uses 2-groups. M-theory in the guise of 11-dimensional supergravity also describes the motion of 2-branes, and Urs Schreiber has noticed that it involves a 3-group - or actually a "3-supergroup", since those darned string theorists are addicted to supersymmetry.

    The only spin foam models we understand really well are the spin foam models for BF theory - I've been harping on that theme for a long time now. So, it's extra fascinating that the BF equations are a special case of the "fake flatness condition" needed in higher gauge theory to get a consistent way to move 1-dimensional entities around.

    I'm not sure what this means yet, but it suggests that 2-groups are lurking around... which is why I got so excited when it looked like Baratin and Freidel's spin foam model for ordinary particle physics was related to the Poincare 2-group spin foam model of Crane and Sheppeard. Right now I'm having a lot of trouble seeing if they really are related...

    Indeed, there's lots of mystery in this subject - lots of great puzzles for smart young folks to tackle!

    That's why my grad students Derek Wise and Jeff Morton are up at the Perimeter Institute now, talking with Aristide Baratin and other people. They know about 2-categories and they know about physics; they should be able to figure something out....

    I've also been working with Jeff and Derek on point particles, strings etc. as "punctures" in space - this could be a way to get BF theory to do something really interesting, and it's also highly related to the n-categorical idea of "extended TQFTs".

    So, there are lots of puzzle pieces, and not enough hours in the day to fit them together. Indeed all this stuff I'm talking about here is just a tiny fraction of what I'd like to understand.

    My own curiosity is also unbridled; unfortunately nobody knows the answers to most of my questions.
     
  12. Jun 27, 2006 #11

    marcus

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    the last two posts #9 and 10 are really really good
    thanks
    I am going to print them out and go study
     
  13. Jul 3, 2006 #12

    marcus

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    I don't mean to get off topic: this thread is about the Baez Schreiber paper on higher gauge theory.
    But as a sidelight on JB's coauthor Urs Schreiber I simply want to note his remarkable blog-reporting of a seminar in the math department at University of Hamburg. This shares the seminar via web with several others who come to "String Coffee Table" blog to take part. It speaks well for Urs as a motivated teacher and web-journalist, as well as mathematician/physicist.

    Seminar on 2-Vector Bundles and Elliptic Cohomology,
    Part I
    http://golem.ph.utexas.edu/string/archives/000737.html
    Part II
    http://golem.ph.utexas.edu/string/archives/000738.html
    Part III
    http://golem.ph.utexas.edu/string/archives/000740.html
    Part IV
    http://golem.ph.utexas.edu/string/archives/000741.html
    Part V
    http://golem.ph.utexas.edu/string/archives/000862.html
     
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